N1-0103 Commuting varieties and Hilbert schemes of points

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Research project is (co) funded by the Slovenian Research Agency.

UL Member: Faculty of Mathematics and Physics
Code: N1-0103
Project: Commuting varieties and Hilbert schemes of points
Period: 1.10.2019 - 30.9.2021   
Range per year: 1,47 FTE, category A
Head: Klemen Šivic
Research activity: Natural sciences and mathematics
Partner Research Organisations: -
Citations for bibliographic records

Project description:

The purpose of the classical algebraic geometry is the study of varieties, which are defined as solutions of systems of polynomial equations. However, in the middle of the previous century it became evident that varieties are insufficient to describe all the geometry, therefore a generalization of a variety is needed. Such generalization is a scheme, which, for example, takes into account multiple solutions or enables the study of systems of equations over any commutative rings, not only polynomials.

Hilbert schemes are fundamental geometric objects which parameterize certain subschemes of a given scheme. They were introduced over 50 years ago by Grothendieck during his development of the modern algebraic geometry. Despite having been investigated since then, they are still not well understood, even in the simplest case of Hilbert schemes of points in affine spaces. Even the very basic question when such scheme is reducible (i.e. when it can be written as a union of smaller schemes) is not solved completely. Moreover, in reducible cases very few irreducible components are known. Our main goal is to investigate them through the connection to varieties of commuting matrices, where the same questions are open. We plan to answer the question of reducibility of commuting varieties and Hilbert schemes in some open cases and to find new irreducible components in the reducible cases. In particular, we intend to characterize the irreducible components of varieties of commuting matrices of small sizes and of Hilbert schemes of few points. Our results are expected to be a basis for future characterization of irreducible components of commuting varieties and Hilbert schemes of points.

Although the connection between varieties of commuting matrices and Hilbert schemes of points in affine spaces seems to have been known earlier, it has only been described in a this year's paper for more than two-dimensional affine space. Moreover, this connection has never been used, except for a direct application of the PI's results on varieties of triples of commuting matrices to Hilbert schemes. This project will pioneeringly fully exploit this connection, investigating commuting varieties and Hilbert schemes simultaneously. Methodologically, techniques from linear, commutative and Lie algebra and algebraic geometry will be used.

Our results will directly impact the fields of linear, commutative and Lie algebra, as well as neighbouring scientific fields. Commuting varieties have a strong impact on modular representation theory. Both, commuting matrices and Hilbert schemes, are important for the questions about ranks of tensors, which have important applications in engineering and statistics.